scholarly journals Cohomology of BiHom-associative algebras

2020 ◽  
pp. 2250008
Author(s):  
Apurba Das
Keyword(s):  
Hochschild Cohomology ◽  
Cochain Complex ◽  
Associative Algebras ◽  
Homotopy Algebras

Bihom-associative algebras have been recently introduced in the study of group hom-categories. In this paper, we introduce a Hochschild type cohomology for bihom-associative algebras with suitable coefficients. The underlying cochain complex (with coefficients in itself) can be given the structure of an operad with a multiplication. Hence, the cohomology inherits a Gerstenhaber structure. We show that this cohomology also control corresponding formal deformations. Finally, we introduce bihom-associative algebras up to homotopy and show that some particular classes of these homotopy algebras are related to the above Hochschild cohomology.

On a Lie Algebraic Approach to Abelian Extensions of Associative Algebras

2020 ◽  
pp. 1-14
Author(s):  
Youjun Tan ◽  
Senrong Xu
Keyword(s):  
Exact Sequence ◽  
Short Exact Sequence ◽  
Cohomology Group ◽  
Hochschild Cohomology ◽  
Algebraic Approach ◽  
Abelian Extension ◽  
Associative Algebras ◽  
Abelian Extensions ◽  
Hochschild Cohomology Group ◽  
Obstruction Class

Abstract By using a representation of a Lie algebra on the second Hochschild cohomology group, we construct an obstruction class to extensibility of derivations and a short exact sequence of Wells type for an abelian extension of an associative algebra.

scholarly journals HIGHER STRING TOPOLOGY ON GENERAL SPACES

2006 ◽  
Vol 93 (2) ◽  
pp. 515-544 ◽  
Author(s):  
PO HU
Keyword(s):  
Simplicial Complex ◽  
Hochschild Cohomology ◽  
Chain Complex ◽  
Derived Category ◽  
Cochain Complex ◽  
String Topology ◽  
Sphere Bundle ◽  
Chain Complexes ◽  
Natural Notion ◽  
Finite Simplicial Complex

In this paper, I give a generalized analogue of the string topology results of Chas and Sullivan, and of Cohen and Jones. For a finite simplicial complex $X$ and $k \geq 1$, I construct a spectrum $Maps(S^k, X)^{S(X)}$, which is obtained by taking a generalization of the Spivak bundle on $X$ (which however is not a stable sphere bundle unless $X$ is a Poincaré space), pulling back to $Maps(S^k, X)$ and quotienting out the section at infinity. I show that the corresponding chain complex is naturally homotopy equivalent to an algebra over the $(k + 1)$-dimensional unframed little disk operad $\mathcal{C}_{k + 1}$. I also prove a conjecture of Kontsevich, which states that the Quillen cohomology of a based $\mathcal{C}_k$-algebra (in the category of chain complexes) is equivalent to a shift of its Hochschild cohomology, as well as prove that the operad $C_{\ast}\mathcal{C}_k$ is Koszul-dual to itself up to a shift in the derived category. This gives one a natural notion of (derived) Koszul dual $C_{\ast}\mathcal{C}_k$-algebras. I show that the cochain complex of $X$ and the chain complex of $\Omega^k X$ are Koszul dual to each other as $C_{\ast}\mathcal{C}_k$-algebras, and that the chain complex of $Maps(S^k, X)^{S(X)}$ is naturally equivalent to their (equivalent) Hochschild cohomology in the category of $C_{\ast}\mathcal{C}_k$-algebras.

scholarly journals Derivations of Murray-von Neumann Algebras

2014 ◽  
Vol 115 (2) ◽  
pp. 206 ◽  
Author(s):  
Richard V. Kadison ◽  
Zhe Liu
Keyword(s):  
Von Neumann Algebra ◽  
Hochschild Cohomology ◽  
Von Neumann Algebras ◽  
Associative Algebras ◽  
Von Neumann ◽  
Finite Von Neumann Algebra ◽  
Algebra Of Operators

A Murray-von Neumann algebra is the algebra of operators affiliated with a finite von Neumann algebra. In this article, we study derivations of Murray-von Neumann algebras and their properties. We show that the "extended derivations" of a Murray-von Neumann algebra, those that map the associated finite von Neumann algebra into itself, are inner. In particular, we prove that the only derivation that maps a Murray-von Neumann algebra associated with a von Neumann algebra of type ${\rm II}_1$ into that von Neumann algebra is 0. This result is an extension, in two ways, of Singer's seminal result answering a question of Kaplansky, as applied to von Neumann algebras: the algebra may be non-commutative and contain unbounded elements. In another sense, as we indicate in the introduction, all the derivation results including ours extend what Singer's result says, that the derivation is the 0-mapping, numerically in our main theorem and cohomologically in our theorem on extended derivations. The cohomology in this case is the Hochschild cohomology for associative algebras.

scholarly journals ON THE HOCHSCHILD HOMOLOGY OF INVOLUTIVE ALGEBRAS

2017 ◽  
Vol 60 (1) ◽  
pp. 187-198
Author(s):  
RAMSÈS FERNÀNDEZ-VALÈNCIA ◽  
JEFFREY GIANSIRACUSA
Keyword(s):  
Hochschild Cohomology ◽  
Homological Algebra ◽  
Homology Theory ◽  
Hochschild Homology ◽  
Associative Algebras ◽  
Derived Functor ◽  
Definition Of

AbstractWe study the homological algebra of bimodules over involutive associative algebras. We show that Braun's definition of involutive Hochschild cohomology in terms of the complex of involution-preserving derivations is indeed computing a derived functor: the ℤ/2-invariants intersected with the centre. We then introduce the corresponding involutive Hochschild homology theory and describe it as the derived functor of the pushout of ℤ/2-coinvariants and abelianization.

scholarly journals Equivariant one-parameter deformations of associative algebras

2019 ◽  
Vol 19 (06) ◽  
pp. 2050114 ◽  
Author(s):  
Goutam Mukherjee ◽  
Raj Bhawan Yadav
Keyword(s):  
Finite Group ◽  
Hochschild Cohomology ◽  
Group Actions ◽  
Associative Algebras ◽  
Finite Group Actions ◽  
Equivariant Version

We introduce an equivariant version of Hochschild cohomology as the deformation cohomology to study equivariant deformations of associative algebras equipped with finite group actions.

scholarly journals Hochschild Cohomology, Monoid Objects and Monoidal Categories

2020 ◽  
Author(s):  
Magnus Hellstrøm-Finnsen
Keyword(s):  
Hochschild Cohomology ◽  
Abelian Groups ◽  
Cochain Complex ◽  
Monoidal Categories ◽  
Equivalent Formulation ◽  
Theoretical Formulation ◽  
Cohomology Monoid

Abstract This paper expands further on a category theoretical formulation of Hochschild cohomology for monoid objects in monoidal categories enriched over abelian groups, which has been studied in Hellstrøm-Finnsen (Commun Algebra 46(12):5202–5233, 2018). This topic was also presented at ISCRA, Isfahan, Iran, April 2019. The present paper aims to provide a more intuitive formulation of the Hochschild cochain complex and extend the definition to Hochschild cohomology with values in a bimodule object. In addition, an equivalent formulation of the Hochschild cochain complex in terms of a cosimplicial object in the category of abelian groups is provided.

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